The Implicit Function Theorem and Non-Existence of Limit of Functions of Several Variables
IJMEST Volume 39, Number 3, ISSN 0020-739X
We use the Implicit Function Theorem to establish a result of non-existence of limit to a certain class of functions of several variables. We consider functions given by quotients such that both the numerator and denominator functions are null at the limit point. We show that the non-existence of the limit of such function is related with the gradient vectors of the numerator and denominator functions. We prove the limit does not exist if the dimension of the vector subspace spanned by the gradient vectors is [greater than or equal] 1.
dos Santos, A.L.C. & da Silva, P.N. (2008). The Implicit Function Theorem and Non-Existence of Limit of Functions of Several Variables. International Journal of Mathematical Education in Science and Technology, 39(3), 389-394.